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arXiv:2607.01301

Dirac oscillator in a helically twisted spacetime with axial torsion

Matheus D. Moro, Fabiano M. Andrade, Faizuddin Ahmed, Edilberto O. Silva

hep-thquant-phSubmitted Jul 1, 2026• Announced Jul 2, 2026• Revised Jul 7, 2026 (v2)
v1 · Jul 1, 2026v2 · Jul 7, 2026
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Abstract

We investigate the Dirac oscillator in a helically twisted spacetime endowed with a uniform axial torsion. Starting from an orthonormal coframe, we compute the Levi--Civita spin connection explicitly and separate the geometric contribution from the axial contortion. Retaining the matrix $β$ in the radial Moshinsky coupling, we show that the second-order problem is the ordered product $\hatΠ_+\hatΠ_-$ rather than the square of a single operator. The resulting radial dynamics is a coupled, self-adjoint two-component system in which the spin connection supplies the correct cylindrical radial operator, while the off-diagonal metric generates the helical combination $m/r-ωk$ and a Coulomb-like geometric term. A finite-element solution reproduces the planar Dirac-oscillator spectrum in the flat limit and reveals asymmetric dependence on the longitudinal momentum, avoided level crossings, and a supersymmetric zero mode at $E=Mc^2$. The axial torsion and longitudinal momentum preserve this zero mode, whereas the helical twist lifts it quadratically. Sector-resolved thermodynamic functions are obtained from the relativistic bound-state spectrum. The explicit spinors further determine longitudinal vector and axial currents, and a Witten-index analysis identifies the helical twist as the deformation that removes the protected zero mode.

Comments
19 pages, 7 figures, improved introduction and corrected references

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We investigate the Dirac oscillator in a helically twisted spacetime endowed with a uniform axial torsion. Starting from an orthonormal coframe, we compute the Levi--Civita spin connection explicitly and separate the geometric contribution from the axial contortion. Retaining the matrix βββ in the radial Moshinsky coupling, we show that the second-order problem is the ordered product Π^+Π^−\hatΠ_+\hatΠ_-Π^+​Π^−​ rather than the square of a single operator. The resulting radial dynamics is a coupled, self-adjoint two-component system in which the spin connection supplies the correct cylindrical radial operator, while the off-diagonal metric generates the helical combination m/r−ωkm/r-ωkm/r−ωk and a Coulomb-like geometric term. A finite-element solution reproduces the planar Dirac-oscillator spectrum in the flat limit and reveals asymmetric dependence on the longitudinal momentum, avoided level crossings, and a supersymmetric zero mode at E=Mc2E=Mc^2E=Mc2. The axial torsion and longitudinal momentum preserve this zero mode, whereas the helical twist lifts it quadratically. Sector-resolved thermodynamic functions are obtained from the relativistic bound-state spectrum. The explicit spinors further determine longitudinal vector and axial currents, and a Witten-index analysis identifies the helical twist as the deformation that removes the protected zero mode.
We investigate bound states of a non-relativistic scalar particle in a three-dimensional helically twisted (torsional) geometry, considering both the free case and the presence of external radial interactions. The dynamics is described by the Schrödinger equation on a curved spatial background and, when included, by minimal coupling to a magnetic vector potential incorporating an Aharonov--Bohm flux. After separation of variables, the problem reduces to a one-dimensional radial eigenvalue equation governed by an effective potential that combines torsion-induced Coulomb-like and centrifugal-like structures with magnetic/flux-dependent terms and optional model interactions. Because closed-form analytic solutions are not reliable over the parameter ranges required for systematic scans, we compute spectra and eigenfunctions numerically by formulating the radial equation as a self-adjoint Sturm--Liouville problem and solving it with a finite-difference discretization on a truncated radial domain, with explicit convergence control. We analyze four representative scenarios: (i) no external potential, (ii) Cornell-type confinement, (iii) Kratzer-type interaction, and (iv) the small-oscillation regime around the minimum of a Morse potential. We present systematic trends of the low-lying levels as functions of the torsion parameter, magnetic field, and azimuthal sector, and we show that geometric couplings alone can produce effective confinement even in the absence of an external interaction.
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We show that a propagating electron wave in the lowest cylindrical channel carries a spin-resolved helical conserved current, and connect this current texture to its local chiral coupling. In a weakly relativistic positive-energy Dirac projection, the two sectors have the same azimuthally symmetric charge density and opposite circulation superposed on longitudinal flow, even though their nonrelativistic orbital harmonic is l=0l=0l=0. The current is radially distributed and extends into the evanescent tail. A scalar screw potential couples these modes through a derived vertex expressed by the \emph{chiral current overlap}, Jχ\mathcal J_χJχ​. Both spatial textures determine the coupling: the environmental radial profile weights the electronic circulation, while the screw harmonic selects correlated changes of spin sector and longitudinal momentum. For a uniform static harmonic, elastic energy conservation makes the first-order spin-flip amplitude vanish within the same transverse mode. Environmental energy exchange is therefore required to activate these selected channels. The same current overlap supplies the spatial kernel for heliciton-assisted inelastic scattering in a companion study and for a subsequent transport treatment of chirality-induced spin selectivity (CISS).
We show that a propagating Dirac electron with intrinsic spin generically carries a real--space helical conserved current, even in the absence of orbital angular momentum. Using exact Dirac eigenstates in cylindrical confinement, we demonstrate that this helical structure possesses definite handedness, persists into evanescent regions, and is characterized by a geometric helix pitch independent of the longitudinal de~Broglie wavelength. This intrinsic helical geometry enables a local geometric coupling between a propagating electron and a chiral environment, yielding chirality--dependent spin selectivity through current geometry rather than through a spin--orbit coupling term.
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We investigate the scattering of a charged quantum particle in a helically twisted background that induces an effective Coulomb-like interaction, in the presence of an Aharonov-Bohm (AB) flux. Starting from the nonrelativistic Schrödinger equation in the twisted metric, we derive the radial equation and show that, after including the AB potential, it can be mapped onto the same Kummer-type differential equation that governs the planar 2D2D2D Coulomb +++ AB problem, with a geometry-induced Coulomb strength and the azimuthal quantum number shifted as m→m−λm\to m-λm→m−λ. We construct the exact scattering solutions, obtain closed expressions for the partial-wave SSS matrix and phase shifts, and derive the corresponding scattering amplitude, differential cross section, and total cross section. We also show that the pole structure of the SSS matrix is consistent with the bound-state quantization previously obtained for the helically twisted Coulomb-like problem.